Machine learning offers an exciting opportunity to improve the calibration of
nearly all reconstructed objects in high-energy physics detectors. However,
machine learning approaches often depend on the spectra of examples used during
training, an issue known as prior dependence. This is an undesirable property
of a calibration, which needs to be applicable in a variety of environments.
The purpose of this paper is to explicitly highlight the prior dependence of
some machine learning-based calibration strategies. We demonstrate how some
recent proposals for both simulation-based and data-based calibrations inherit
properties of the sample used for training, which can result in biases for
downstream analyses. In the case of simulation-based calibration, we argue that
our recently proposed Gaussian Ansatz approach can avoid some of the pitfalls
of prior dependence, whereas prior-independent data-based calibration remains
an open problem.
Bias and Priors in Machine Learning Calibrations for High Energy Physics
高エネルギー物理のための機械学習校正のバイアスと先行
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Rikab Gambhir,1, 2, ∗ Benjamin Nachman,3, 4, † and Jesse Thaler1, 2, ‡
Rikab Gambhir,1, 2, ∗ Benjamin Nachman,3, 4, , and Jesse Thaler1, 2, .
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MIT-CTP 5432
MIT-CTP 5432
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1Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
マサチューセッツ工科大学理論物理学センター, ケンブリッジ, ma 02139, usa
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2The NSF AI Institute for Artificial Intelligence and Fundamental Interactions
第2回 NSF AI Institute for Artificial Intelligence and Basic Interactions
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3Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
カリフォルニア大学バークレー校ローレンス・バークレー国立研究所 3Physics Division, CA 94720, USA
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4Berkeley Institute for Data Science, University of California, Berkeley, CA 94720, USA Machine learning offers an exciting opportunity to improve the calibration of nearly all reconstructed objects in high-energy physics detectors.
カリフォルニア大学バークレー校の4Berkeley Institute for Data Science, University of California, Berkeley, CA 94720, USA Machine Learningは、高エネルギー物理検出器のほとんどすべての再構成されたオブジェクトの校正を改善するエキサイティングな機会を提供する。 訳抜け防止モード: 4Berkeley Institute for Data Science, University of California, Berkeley, CA 94720 USA Machine Learningはエキサイティングな機会を提供する 高エネルギー物理検出器におけるほとんどすべての再構成された物体の校正を改善する。
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However, machine learning approaches often depend on the spectra of examples used during training, an issue known as prior dependence.
This is an undesirable property of a calibration, which needs to be applicable in a variety of environments.
これはキャリブレーションの望ましくない性質であり、様々な環境に適用する必要がある。
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The purpose of this paper is to explicitly highlight the prior dependence of some machine learning-based calibration strategies.
本研究の目的は,機械学習による校正戦略の事前依存性を明確にすることである。
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We demonstrate how some recent proposals for both simulation-based and data-based calibrations inherit properties of the sample used for training, which can result in biases for downstream analyses.
In the case of simulation-based calibration, we argue that our recently proposed Gaussian Ansatz approach can avoid some of the pitfalls of prior dependence, whereas prior-independent data-based calibration remains an open problem.
A. Simulation-based Calibration B. Prior Dependence and Bias C. Mitigating Prior Dependence D. Data-based Calibration E. Unbiased Data-based Approaches?
a. シミュレーションに基づく校正 b. 事前依存とバイアス c. 事前依存の緩和 d. データベース校正 e. unbiased data-based approach?
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III. Resolution and Uncertainty in Calibrations
III。 校正の解決と不確かさ
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A. Resolution B. Uncertainty
a.解決b.不確実性
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IV. Gaussian Examples A. Simulation-based Calibration B. Data-based Calibration
IV。 ガウスの例 A. シミュレーションに基づく校正B. データに基づく校正
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V. Calibrating Jet Energy Response
V. ジェットエネルギー応答の校正
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A. Datasets B. Simulation-based Calibration C. Data-based Calibration
a. データセット b. シミュレーションベースキャリブレーション c. データベースキャリブレーション
Calibration is the task of removing bias from an inference – that is, to ensure the inference is “correct on average”.
校正は、推論からバイアスを取り除くタスクです。つまり、推論が“平均的に正しい”ことを保証します。
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The are two major classes of calibration: simulation-based calibration, where the goal is to infer a truth reference object, and data-based calibration, where the goal is to match simulation and data distributions.
Both simulation-based calibrations and data-based calibrations are essential components of the experimental program in high-energy physics (HEP), and a significant amount of time is spent deriving these results to enable downstream analyses.
We focus on the ATLAS and CMS experiments at the Large Hadron Collider (LHC) for our examples, but this discussion is relevant for all of HEP (and really any experiment).
ATLAS and CMS have performed many recent calibrations, including the energy calibration of single hadrons [1, 2], jets [3, 4], muons [5, 6], electrons/photons [7–9], and τ leptons [10, 11].
The reconstruction efficiencies of all of these objects are also calibrated and include the classification efficiency of jets from heavy flavor [12, 13] and even more massive particles [14, 15].
Machine learning is a promising tool to improve both types of calibration.
機械学習は両方のキャリブレーションを改善するための有望なツールだ。
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In particular, machine learning methods can readily process high-dimensional inputs and therefore can incorporate more information to improve the precision and accuracy of a calibration.
There have been a large number of proposals for improving the simulationbased calibrations of various object energies, including single hadrons [16–21], muons [22], and jets [23–33] at colliders; kinematic reconstruction in deep inelastic scattering [34]; and neutrino energies in a variety of experiments [35–40].
A non-universal calibration would have a rather limited utility, and can produce undesirable results if applied to a dataset that does not exactly match the calibration dataset.
A calibration can be biased due to the choice of estimator or fitting procedure used, even if the usual pitfalls of dataset-induced biases are taken care of.
In this paper, we explain the origin of prior dependence for common calibration techniques, with explicit illustrative examples, and demonstrate the associated bias that these procedures incur.
For simulation-based calibrations, we advocate for our Gaussian Ansatz [43] as a machine-learning-bas ed strategy that is prior independent and bias-free.
For data-based calibrations, we are unaware of any prior-independent methods in the literature.
データに基づくキャリブレーションでは、文献に先行する非依存の手法を知らない。
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Many of the conclusions of this paper are well-known to the experts, but we hope that by highlighting these issues, we can inspire the development of prior-independent calibration methods.
II. THE STATISTICS OF CALIBRATION In this section, we review some of the basic features of simulated-based and data-based calibration, and discuss the issues of prior dependence and bias.
2 Prior independence is a necessary prerequisite for closure.
2 事前独立は閉鎖に必要な前提条件である。
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How- ever, even with prior independence, closure is not guaranteed.
方法 たとえ 事前の独立であっても 閉鎖は保証されない
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2 A. Simulation-based Calibration
2 A. シミュレーションに基づく校正
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In simulation-based calibration, the goal is to infer target (or true) features zT ∈ RN from detector-level features xD ∈ RM – that is, to construct an estimator or calibration function f : RM → RN where
To carry out simulation-based calibration, one starts with a set of (xD, zT ) pairs, which typically come from an in-depth numerical simulation of an experiment.
V, xD will be the experimentally measurable features of hadronic jets and zT will be the true jet energy.
V, xDはハドロンジェットの実験的に測定可能な特徴であり、zTは真のジェットエネルギーとなる。
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For concreteness, one can think of the calibration function f as being parameterized by a universal function approximator such as a neural network, whose weights and biases are learned.
This is often done by minimizing the mean squared error (MSE) loss:
これは、平均二乗誤差(MSE)損失を最小限にすることで行われることが多い。
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fMSE = argmax
fMSE = argmax
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g Etrain[(g(XD) − ZT )2],
g Etrain[(g(XD) − ZT )2],
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(2) where capital letters correspond to random variables and E represents the expectation value over the training sample used to derive the calibration.
Using the calculus of variations, one can show that with enough training data, a flexible enough functional parameterization, and a sufficiently exhaustive training procedure, the asymptotic solution to Eq (2) is:
(3) where lowercase letters correspond to an instance of a random variable.
(3) 小文字がランダム変数のインスタンスに対応する場合。
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In this way, f learns the mean value of zT for a given xD in the training set.
このようにして、f はトレーニングセット内の与えられた xD に対する zT の平均値を学ぶ。
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Alternative loss functions result in statistics other than the mean.
別の損失関数は平均以外の統計結果をもたらす。
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See e g Ref.
e g ref を参照。
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[44] for alternative approaches, including mode learning, which is a standard target for many traditional calibrations (usually in the form of truncated Gaussian fits; see e g [9]).
[44] は、多くの伝統的なキャリブレーションの標準的ターゲットであるモード学習を含む代替のアプローチである(通常、ガウス整合の形式である: e g [9] を参照)。
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B. Prior Dependence and Bias
B.事前依存とバイアス
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A key assumption of simulation-based calibration is
シミュレーションに基づく校正の鍵となる仮定は
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that the detector response is universal:
検出器の応答は普遍的です
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ptest(xD|zT ) = ptrain(xD|zT ).
ptest(xD|zT ) = ptrain(xD|zT )。
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(4) This equation says that for a given truth input zT , the detector response is the same between the training data used for deriving the calibration and the testing data used for deploying the calibration.
Note that even if the training dataset is statistically identical to the testing dataset (i.e. ptest(zT ) = ptrain(zT )), it is not guaranteed that the calibration will be unbiased.
One way to reduce the bias is if the prior is “wide and flat enough”, such that the prior asymptotically approaches a uniform sampling over the real line relative to the detector response.
(8) that if the prior p(zT ) is Gaussian with extent σ, and the detector response p(xD|zT ) is a Gaussian noise model with extent , then the bias scales as:
(12) where MLC stands for maximum likelihood classifier – see Ref.
(12) MLC は最大可算分類器の略で、Ref を参照。
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[48]. Again, because the detector response p(xD|zT ) is universal, maximum likelihood calibrations are universal,3 and in certain configurations, are provably unbiased.
[43] is to estimate the (local) likelihood density by extremizing the Donsker-Varadhan representation (DVR) [49, 50] of the Kullback-Leibler divergence [51]:
(14) By parametrizing f(xD, zT ) via a specially chosen Gaussian Ansatz (see Ref. [43] for details), one can extract the local maximum likelihood estimate and resolution with a single neural network training.
If the fraction of signal is different in the training set and the test set, that is, ptest(zT ) 6= ptrain(zT ), then the output can no longer be interpreted as the probability of the signal.
Luckily, classifiers (often called taggers) are almost never used this way in HEP, since the classification score is not interpreted directly as a probability.5
In data-based calibration, the goal is to account for possible differences between a true detector response, pdata(xD) and a simulated detector model psim(xD).
That is, the goal is to match detector level features xD between data and a simulation at the distribution level, in contrast to simulation-based distribution, where the goal is to match xD and a target feature zT at the object level.
In the machine learning literature, data-based calibration is called domain adaptation.
機械学習の文献では、データに基づくキャリブレーションをドメイン適応と呼ぶ。
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Machine learning domain adaptation has been widely studied in the context
機械学習の領域適応は、文脈で広く研究されてきた
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psim(xD) = R dzT psim(xD|zT ) ptrain(zT ) is a simulated
psim(xd) = r dzt psim(xd|zt ) ptrain(zt ) はシミュレーションである。
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4 It is not always true that a maximum likelihood calibration is unbiased.
4 最大極大校正が不偏であることは必ずしも事実ではない。
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For instance, if XD is drawn from a uniform distribution U(0, zT ), then the maximum likelihood estimate from a single xD sample is ˆzT = xD, whereas an unbiased estimate would be ˆzT = 2xD.
As we will see, though, there is implicit prior dependence in h.
しかし、私たちが見るように、h には暗黙の事前依存がある。
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For simplicity, consider the special case of one dimension.
単純さのため、1次元の特別な場合を考える。
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Here, for any OT metric, the OT map h : R → R is simply given by:
ここで、任意の OT 計量に対して、OT 写像 h : R → R は次のように与えられる。
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h(xD) = P −1
h(xD) = P−1
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Pλ(xD) =R xD−∞ dx0
Pλ(xD) = R xD−∞ dx0
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(17) where Pλ is the cumulative distribution function of λ, i.e. ).
17) ここで pλ は λ,すなわち λ の累積分布関数である。
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This function maps quantiles of the simulated distribution to quantiles of the data distribution.
この関数は、シミュレーション分布の分位数をデータ分布の分位数にマップする。
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The Jacobian of this transformation is:
この変換のヤコビアン(Jacobian)は、
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data(Psim(xD)),
データ(psim(xd))
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D pλ(x0 D
D pλ(x0) D
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|h0(xD)| = psim(xD) pdata(h(xD))
|h0(xD)| = psim(xD) pdata(h(xD))
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R dzT psim(xD|zT ) ptrain(zT )
R dzT psim(xD|zT ) ptrain(zT )
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= pdata(h(xD))
= pdata(h(xD))
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(18) . Thus, since the prior ptrain(zT ) explicitly appears, the derived OT-based detector model in Eq (16) is prior dependent.
(18) . したがって、前のptrain(zT ) が明示的に現れるため、Eq (16) における OT ベースの検出器モデルが事前に依存している。
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In line with simulation-based calibration, the bias of a data-based calibration is the average difference between the estimator ˆp(xD) and the desired value pdata(xD), conditioned on xT .7 For OT-based calibration, the bias per
7 This differs from the simulation-based calibration definition, which was conditioned on zT .
7) シミュレーションに基づくキャリブレーション定義と異なり,zt 上で条件づけされた。
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In data, there is no truth level zT .
データでは、真理レベル zT はありません。
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However, sometimes, a proxy can be used as a zT in data, allowing for a direct comparison of true versus reconstructed zT values in data-based calibration.
At least in the case the special case of one dimensional OT-based calibration, however, we have shown above that the corrected response function is not universal.
This implies that all databased calibration methods in use are biased, though the degree of bias may be small if the testing and training truth-level densities are similar enough.
A. Resolution As already mentioned, the bias of a calibration refers to the difference in central tendency (such as the mean, median, or mode) between a reconstructed quantity and a reference quantity.
a. 解決 キャリブレーションのバイアス(英: bias of a calibration)とは、再構成された量と基準量の間の中央傾向(平均、中央値、モードなど)の違いを指す。
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By contrast, the resolution of a calibration refers to the spread in the difference between the reconstructed and reference quantities.
対照的に、校正の解像度は、再構成された量と基準量の差の広がりを指す。
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Using variance as our measure of spread, the resolution Σ2(zT ) can be
分散を拡散の尺度として使うと、分解能 σ2(zt ) は
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Σ2(zT ) = Vartest[f(XD) − zT|ZT = zT ].
σ2(zt ) = vartest[f(xd) − zt|zt = zt ] である。
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(20) Resolutions, like biases, can be prior-dependent.
The prior-dependence is seen by applying Bayes’ Theorem to ptrain(z0 As before, this prior dependence can be reduced if the prior is wide compared to the detector response.
If the prior p(zT ) is Gaussian with extent σ, and the detector response p(xD|zT ) is a Gaussian noise model with extent , then by applying Eq (22), one can show that the resolution scales as:
σ On the other hand, for the prior-independent MLC calibration (Eq.
σ 一方,事前独立型mlcキャリブレーション(eq)について検討した。
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(12)), the resolution can be shown to be:
(12)) 解像度は次のようになる。
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Σ2(zT ) = 2.
σ2(zt ) = σ2 である。
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(23) In HEP (and many other) applications, however, it is common to instead refer to the resolution with respect to a measurement xD rather than the true value zT .
That is, for an inference ˆzT = f(xD), we would like a measure of the spread of zT values consistent with this measurement, which we will denote Σ(xD) (distinguished by the xD argument rather than zT ).
Depending on the context and type of calibration, there are a variety of ways to define Σ(xD) – for instance, as the standard deviation from a Gaussian fit to the distribution of reconstructed over true energies (see e g Ref. [45]).
However, for frequentist approaches where the posterior is not well defined, such as the maximum likelihood calibration, the resolution cannot be defined this way and care must be taken.
For Gaussian noise models p(xD|zT ), the likelihood is symmetric under interchanging the arguments xD and zT , so one can take the resolution to be (applying Eq (20)): Σ2(xD) = Σ2(zT ) = 2.
example, if a calibration requires multiplying the reconstructed quantity by a fixed number greater than one, then the resolution will grow by the same amount.
For instance, in the context of jet energy calibrations, xD = α η for some constant α and an observable quantity η (e g energy dependence on the pseudorapidity).
例えば、ジェットエネルギーのキャリブレーションの文脈では、ある定数 α に対して xD = α η であり、観測可能な量 η である(例えば、擬レイピディティへのエネルギー依存)。
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If any of the ~yD have a non-trivial probability density, this will be inherited by the reconstructed value xD and thus xD will have a non-zero resolution.
This resolution is completely reducible, however, through a calibration that is ~yD-dependent – that is, a calibration function ˆzT = f0(~yD) rather than ˆzT = f(xD).
The ability to incorporate many auxiliary features is why machine-learning-bas ed approaches, such as the Gaussian Ansatz [43], have the potential to improve analyses at HEP experiments.
B. Uncertainty In the machine learning literature, “resolution” would be referred to as a type of “uncertainty”.
B.不確かさ 機械学習の文献では、“解決”は“確実性”の一種として言及される。
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Uncertainty in the statistical context refers to the limited information about zT contained in xD.
統計的文脈の不確かさは、xDに含まれるzTに関する限られた情報を指す。
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In the HEP literature, though, we use uncertainty in a different way, to instead refer to the limited information we have about the bias and resolution of a calibration.
The reason for this difference in nomenclature is that HEP research is based primarily on simulation-based inference, where data are analyzed by comparison to model predictions.
A worse resolution can degrade the statistical precision of a measurement, but if it is well-modeled by the simulation, then there is no associated systematic uncertainty (though there will still be statistical uncertainties).
The mo- 6 mentum inbalance between the jet and the Z boson will be due in part to differences in the calibration between data and simulation and in part due to the mismodeling of initial and final state radiation.
IV. GAUSSIAN EXAMPLES In this section, we demonstrate some of the calibration issues related to bias and prior dependence in a simple Gaussian example.
We assume that the truth information (the “prior”) is distributed according to a Gaussian distribution with mean µ and variance σ2:
真理情報(「主」)は平均 μ と分散 σ2 を持つガウス分布に従って分布していると仮定する。
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ZT ∼ N (µ, σ2).
ZT は N (μ, σ2) である。
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XD|ZT = zT ∼ N (zT , 2).
XD|ZT = zT > N (zT , >2) である。
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(26) The detector response is assumed to induce Gaussian smearing centered on the truth input with variance 2: (27) For the simulation-based calibration in Sec.
IV A, the goal is to learn ZT given XD, assuming perfect knowledge of the detector response.
IVA の目標は、検出器応答の完全な知識を前提として、XD の ZT を学習することである。
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For the data-based calibration in Sec.
secのデータに基づく校正のために。
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IV B, the goal is to map XD in “simulation” to XD in “data”.
IV Bの目的は、XDを“シミュレーション”で“データ”でXDにマッピングすることだ。
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In this latter study, we assume that data and simulation have the same true probability density and differ only in their detector response, sim 6= data – that is, psim “mismodels” pdata.
In Fig 1a, we show the simulated data, for which both the true and reconstructed values follow a Gaussian distribution.
図1aでは、真の値と再構成された値の両方がガウス分布に従うシミュレーションデータを示す。
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The first step of a typical calibration is to predict the true zT from the reconstructed xD.
典型的な校正の最初のステップは、再構成された xD から真の zT を予測することである。
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Since we know that the average dependence of the true zT on the reconstructed xD is linear, we perform a first-order polynomial fit to the data using numpy polyfit, which is represented by the blue dashed line in Fig 1a.
[43], the calibration function B(x) is obtained by minimizing the DVR loss function from Eq (14), such that after training: (36) (37) For Gaussian noise models, this maximum likelihood estimate is unbiased, as confirmed by the numerical results in Fig 1b.
The B and C networks are each a single node with linear activation.
B と C のネットワークは、それぞれ線形活性化を持つ単一ノードである。
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The D network is set to zero by hand.
Dネットワークは手動でゼロに設定される。
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Optimization is carried out with Adam [91] over 100 epochs with a batch size of 128.
最適化はAdam[91]が100エポック以上で行われ、バッチサイズは128である。
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As desired, the Gaussian Ansatz yields a calibration that is independent of the prior ptrain(zT ).
願わくば、ガウスアンザッツは、以前のptrain(zT)とは独立なキャリブレーションを与える。
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to get the bias from the MSE calibration approach: ) p(xD|zT ) p(z0
MSE校正法からバイアスを得る: ) p(xD|zT ) p(z0)
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) p(xD) .
) p(xd) である。
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(38) It is possible to solve Eq (38) analytically for the Gaussian setup:
(38) ガウス集合に対して解析的に Eq (38) を解くことができる。
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To demonstrate the bias, we plug in Eq (28) into Eq (8)
バイアスを示すために、Eq (28) を Eq (8) に差し込む。
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b(zT ) + zT =
b(zT ) + zT =
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dxD dz0 T z0
dxD dz0 T z0
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Z T . T T p(xD|z0 (cid:19)
Z T . T T p(xD|z0 (cid:19)
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(cid:18) 2
(cid:18)~2
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σ2 + 2 b(zT ) =
σ2 + 2 b(zT ) =
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(µ − zT ). (39)
(μ − zT)。 (39)
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7 As expected, b(zT ) → 0 as → 0.
7 予想通り、b(zT ) → 0 は t → 0 である。
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For > 0, though, there is a non-zero bias with the MSE approach.
しかし > 0 の場合、MSE のアプローチには非ゼロバイアスがある。
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The zTbinned resolutions can also be computed using Eqs.
ztbinned resolutionsはeqsを使って計算することもできる。
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(22) and (23): ΣMSE(zD) = σ2 ΣMLC(zD) = .
(22)・(23) ΣMSE(zD) = σ2 などがある。
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2 + σ2 ,
2 + σ2 ,
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(40) (41) The fitted biases and resolutions are presented in Fig 2, which exhibits the bias expected from Eq (39).
(40) (41) 適合バイアスと解像度は図2に示され、eq(39)から期待されるバイアスを示す。
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This illustrates the large bias introduced by the MSE regression procedure.
これは、MSE回帰手順によって導入された大きなバイアスを示している。
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To further highlight the role of prior dependence, we repeat the MSE calibration procedure, where we test multiple values of the prior parameters µ and σ to confirm the predictions in Eq (39).
II E, we are unaware of any priorindependent data-based calibration.
iie 事前の独立データに基づく校正には気付いていない。
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To highlight this challenge, we study the OT-based technique introduced in Ref.
この課題を浮き彫りにするために、Refで導入されたOTベースの技術について研究する。
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[42] and mentioned in Sec.
[42] と Sec に記載されている。
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II D. In our Gaussian example, the goal is to calibrate a “simulation” sample with (µsim., σsim., sim.) to match a “data” sample with (µdata, σdata, data).
For simplicity, we assume that the true spectra (determined by (µ, σ)) are the same in data and in simulation, such that there is no systematic uncertainty in the calibration (see Sec. III B).
単純さのために、真のスペクトル( (μ, σ) によって決定される)はデータとシミュレーションにおいて同じであり、キャリブレーションに系統的不確実性がないと仮定する(sec. iii b を参照)。
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Only , the parameter governing the detector response, is different between simulation and data – the simulation mismodels the real detector.
To highlight the issue of prior-dependence, we consider a “training” set with one value of µtrain = 0 and a “testing” set with a different value of µtest, with a shared value of σ.
(a) 2D Histogram of the reconstructed value xD distribution versus the true value zT distribution, in the Gaussian example with µ = 0, σ = 1, and = 2. The dashed line represents a linear fit to the data points.
(b) For test values of xD, the vertical axis is the calibrated target value ˆzT (xD).
(b)xDの試験値に対して、垂直軸は校正された目標値(zT(xD))である。
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The blue dots are the results from a numerical MSE fit fMSE(xD), and the error bars correspond to the numerical point resolution ΣMSE(xD), with the analytic prediction in the red dotted line.
10 FIG. 4. The data-driven calibration functions corresponding to Fig 3.
10 図4。 データ駆動型校正機能はfig3に対応する。
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The blue points correspond to the calibration function htrain derived from the training set and the red points correspond to the ideal calibration htest one would derive from the test set.
The shaded histograms correspond to the zT = mtrue truth-level distributions, whereas the light triangles and dark circles correspond to xD = mreco for the fast (Delphes) and slow (Geant4) distributions respectively.
A numerical demonstration of this bias is presented in Fig. 3, where histograms of the data and simulation are presented along with the calibrated result.
The actual calibration function is plotted in Fig 4 and compared to the analytic expectation from Eqs.
実際のキャリブレーション関数は図4でプロットされ、Eqsの分析期待値と比較される。
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(44) and (43).
(44)及び(43)
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The fact that the calibration derived on the train set is not the same as the calibration derived on the test set shows that the calibration derived in one and applied to the other will lead to a residual bias.
To illustrate the impact of the prior dependence, we use a realistic and also extreme example where calibrations are derived in a sample of generic quark and gluon jets and then applied to a test sample of jets from the decay of a heavy new resonance.
In practice, jet energy calibrations are derived for individual jets, but this requires at least including calibrating the jet rapidity in addition to the jet energy.
We keep the problem one-dimensional in order to ensure the problem is easy to visualize and to mitigate the dependence on features that are not explicitly modeled.
=0Data-Based Gaussian ExampleTrain Set CalibrationTest Set Calibration1-1180020 00220024002600280030 00mjj [GeV]105104103102101p(mjj ) [GeV1]Dijet DistributionsQCD TruthQCD DelphesQCD GeantBSM TruthBSM DelphesQCD Geant
=0Data-based Gaussian ExampleTrain Set CalibrationTest Set Calibration1-1180020 00220024002600280030 00mjj [GeV]1051041032101p(mjj) [GeV1]Dijet DistributionsQCD TruthQCD DelphesQCD GeantBSM TruthBSM DelphesQCD GeantBSM TruphesQCD Geant
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英語（論文から抽出）
日本語訳
スコア
jj jj simulation sample uses Pythia 6.426 [92] with the Z2 tune [93] and interfaced with a Geant4-based [94–96] full simulation of the CMS experiment [97].
The full simulation sample comes from the CMS Open Data Portal [102–104] and processed into an MIT Open Data format [105–108].
完全なシミュレーションサンプルは、CMS Open Data Portal [102–104] から得られ、MIT Open Dataフォーマット [105–108] に処理されます。
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The fast simulation sample is available at Ref.
高速シミュレーションサンプルはRefで公開されている。
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[109, 110].
[109, 110].
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For each dataset, we have access to the parton-level hard-scattering scale ˆpT from Pythia, which is in general different from the jet-level transverse momentum pT we are interested in studying.
To avoid any issues related to the trigger, we focus on events where ˆpT > 1 TeV.
トリガーに関連する問題を回避するために、 spt > 1 tev のイベントに焦点を当てます。
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Particles (at truth level) or particle flow candidates (at reconstructed level) are used as inputs to jet clustering, implemented using FastJet 3.2.1 [111, 112] and the antikt algorithm [113] with radius parameter R = 0.5.
No calibrations are applied to the reconstructed jets.
復元されたジェットにはキャリブレーションは適用されない。
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In order to emulate two different physics processes while controlling for all hidden variables, we consider dijet events with two different sets of event weights.
Additionally, the mreco distribution is significantly different between the full and fast simulations, which to correct requires a data-based calibration.
The neural network has three hidden layers with 50 nodes per layer, with the rectified linear unit activation for intermediate layers and a linear activation for the output.
Training is performed over the QCD sample to obtain the calibration function.
校正機能を得るために、QCDサンプル上でトレーニングを行う。
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The learned calibration function is then applied to both the QCD and BSM test samples
学習校正関数はQCDおよびBSMテストサンプルの両方に適用される
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The result of MSE calibration is shown in Fig 6a.
MSE校正の結果を図6aに示す。
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Prior to any calibration, the detector response is about 5% low in both the QCD and BSM test samples.
キャリブレーションの前には、検出器の応答はQCDとBSMの両方のテストサンプルで約5%低かった。
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After calibration, the mean is nearly unity for the QCD sample, albeit with a large width – that is to say, the average bias is close to zero over the prior, but the average resolution is large.
The A, B, C, and D networks of the Gaussian Ansatz each consist of three hidden layers with 32 nodes per layer, with the same activation functions, batch size, and epochs as in the Gaussian example.
The calibration function trained on the QCD sample can be used for the BSM sample, and as Fig 6b shows, the calibration is indeed universal and unbiased, as expected.
jj The goal for the data-based calibration task is to “cor), given by the fast simulation (Delphes), rect” psim(mreco ), given by to the observed data distribution pdata(mreco the full simulation (Geant4).
jj データベースのキャリブレーションタスクの目標は、観測されたデータ分散pdata(mreco the full Simulation(Geant4)に与えられる高速なシミュレーション(Delphes, rect)psim(mreco )によって与えられる「正しい」ことである。
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We now apply the same procedure described in Sec.
現在、secで説明されているのと同じ手順を適用する。
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IV B to the dijet example.
IV B を例に挙げる。
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An OT-based calibration is derived using QCD jets, to align the fast simulation Delphes) sample with the full simulation Geant4 sample.
jj 8 The converse is also true – attempting to use a calibration fitted on the BSM sample will lead to bias on the QCD sample, or any other BSM sample for that matter.
On the QCD sample, this calibration closes by construction.
QCDサンプルでは、このキャリブレーションは構築によって閉じる。
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In particular, as shown in Fig 7a, the blue dashed line in the ratio plot fluctuates around unity, with deviations due to statistical fluctuations that differ between the two halves of the event samples.
While the resulting dashed distribution agrees better with the data histogram in dark red than does the fast sim histogram in light red, the overall agreement is still rather poor.
In this paper, we explored the prior dependence of machine learning-based calibration techniques.
本稿では,機械学習に基づくキャリブレーション手法の事前依存性について検討した。
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There is a growing number of machine learning proposals for simulation-based and data-based calibration and in nearly all cases, there is a prior dependence.
We highlighted the resulting calibration bias in a synthetic Gaussian example and a more realistic particle physics example of dijet production at the LHC.
While prior independent, this technique is typically biased and does not scale well to many dimensions.
従来の独立性はあるものの、この手法は一般に偏りがあり、多くの次元にうまくスケールしない。
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We proposed a new approach based on maximum likelihood estimation in Ref.
我々はRefの最大推定値に基づく新しい手法を提案した。
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[43], based on parametrizing the log-likelihood with a Gaussian Ansatz.
[43] ガウスアンザッツによる対数様態のパラメトリゼーションに基づく。
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Maximum-likelihood-b ased approaches are prior independent by construction and are well-motivated statistically.
最大様相に基づくアプローチは、建設によって事前に独立であり、統計的に十分に動機づけられている。
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13 Parametrizing the maximum likelihood estimator with neural networks requires a different learning paradigm than current approaches, but it extends well to many dimensions.
To our knowledge, there are currently no prior-independent data-based calibration approaches.
私たちの知る限り、現在、事前非依存のデータベースのキャリブレーションアプローチはありません。
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To make the most use of the complex data from the LHC and other HEP experiments, it is essential to use all of the available information for object calibration.
This will require modern machine learning to account for all of the subtle correlations in high dimensions.
これは、高次元における微妙な相関をすべて考慮し、現代的な機械学習を必要とする。
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It is important, however, that we construct these machine learning calibration functions in a way that integrates all of the features of classical calibration methods.
We highlighted prior independence in this paper as a cornerstone of calibration.
本稿では,キャリブレーションの基礎として,先行的独立性を強調した。
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In the future, innovations that incorporates knowledge of the detector response or physics symmetries may further enhance the precision and accuracy of machine learning calibrations.
found at https://github.com/h ep-lbdl/calibrationp riors, which makes use of Jupyter notebooks [115] employing NumPy [116] for data manipulation and Matplotlib [117] to produce figures.
All of the machine learning was performed on a Nvidia RTX6000 Graphical Processing Unit (GPU).
すべての機械学習はNvidia RTX6000 Graphical Processing Unit (GPU)上で実行された。
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The physics data sets are hosted on Zenodo at Refs.
物理データセットはzenodo at refsにホストされている。
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[106–108, 110].
[106–108, 110].
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can be ACKNOWLEDGMENTS
できるかもしれない 裏書き
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BN is supported by the U.S. Department of Energy (DOE), Office of Science under contract DE-AC0205CH11231.
BNはD-AC0205CH11231契約の下でアメリカ合衆国エネルギー省(DOE)が支援している。
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RG and JT are supported by the National Science Foundation under Cooperative Agreement PHY-2019786 (The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, http://iaifi. org/), and by the U.S. DOE Office of High Energy Physics under grant number DE-SC0012567.
RGとJTは、協力協定 PHY-20 19786 (NSF AI Institute for Artificial Intelligence and Fundamental Interactions, http://iaifi. org/)の下で国立科学財団によって支援されており、アメリカ合衆国高エネルギー物理学局(DOE Office of High Energy Physics, DE-SC0012567)によって支援されている。
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Patrick T. Komiske, Eric M. Metodiev, Benjamin Nachman, Matthew D. Schwartz, “Pileup Mitigation with Machine Learning (PUMML), JHEP 12, 051 (2017), arXiv:1707.08600 [hep-ph]”。 訳抜け防止モード: (29)Patrick T. Komiske、Eric M. Metodiev、Benjamin Nachman そしてMatthew D. Schwartz氏は,“PUMML(Pileup Mitigation with Machine Learning)”と題する。 JHEP 12, 051 (2017 ), arXiv:1707.08600 [ hep - ph ]
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A. Rogozhnikov, “Reweighting with Boosted Decision Trees” Proceedings, 17th International Workshop on Advanced Computing and Analysis Techniques in Physics Research (ACAT 2016): Valparaiso, Chile, January 18-22, 2016 762, 012036 (2016), arXiv: 1608.05806 [physics.data-an] 訳抜け防止モード: 54 ] A. Rogozhnikov, “ 強化された決定木による再重み付け”。 第17回Advanced Computing and Analysis Techniques in Physics Research(ACAT 2016)に参加して Chile, January 18 - 22, 2016 762, 012036 ( 2016 ), arXiv:1608.05806 [ Physics.data - an ]
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[55] Anders Andreassen and Benjamin Nachman, “Neural Networks for Full Phase-space Reweighting and Parameter Tuning,” Phys.
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15 [56] S. Diefenbacher, E. Eren, G. Kasieczka, A. Korol, B. Nachman, D. Shih, “DCTRGAN: Improving the Precision of Generative Models with Reweighting”. Journal of Instrumentation 15 P11004 (2020), arXiv:2009.03796 [hep-ph]. 訳抜け防止モード: 15 [56 ]S. Diefenbacher, E. Eren, G. Kasieczka, A. Korol, B. Nachman, D. Shih, “DCTRGAN :” 再重み付けによる生成モデルの精度向上」 Journal of Instrumentation 15 P11004 (2020 ), arXiv:2009.03796 [ hep - ph ]
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and Kyle Cranmer, “Learning to Pivot with Adversarial Networks,” in Advances in Neural Information Processing Systems, Vol.
そしてkyle cranmer(カイル・クランマー)氏は、神経情報処理システム(neural information processing systems, vol.)の進歩についてこう述べている。 訳抜け防止モード: そしてKyle Cranmerは語る。 敵ネットワークでPivotを学習する”。 In Advances in Neural Information Processing Systems, Vol .
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[59]James Dolen, Philip Harris, Simone Marzani, Salvatore Rappoccio, Nhan Tran, “Thinking outside the ROCs: Designing Decorrelated Taggers (DDT) for jet substructure), JHEP 05, 156 (2016), arXiv: 1603.00027 [hep-ph]”. 訳抜け防止モード: 59 ] James Dolen, Philip Harris, Simone Marzani, Salvatore RappoccioとNhan Tran : “ROCの外で考える” DDT(Decorrelated Taggers)をジェットサブ構造に設計する」。 JHEP 05, 156 (2016 ), arXiv:1603.00027 [ hep - ph ]
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Christoph Englert氏、Peter Galler氏、Philip Harris氏、Michael Spannowsky氏は、“Machine Learning Uncertainties with Adversarial Neural Networks”と題している。
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そしてjudith m. katzy氏は、"高エネルギー物理学の分類器のサンプルバイアスを減らすための逆領域適応(adversarial domain adaptation)"(2020), arxiv:2005.00568 [stat.ml]。
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[72] Gregor Kasieczka, Benjamin Nachman, Matthew D. Schwartz, and David Shih, “ABCDisCo: Automating the ABCD Method with Machine Learning,” (2020), 10.1103/PhysRevD.103 .035021, arXiv:2007.14400 [hepph].
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